We consider a special class of weak dependent random variables with control on covariances of Lipschitz transformations. This class includes, but is not limited to, positively, negatively associated variables and a few other classes of weakly dependent structures. We prove a Strong Law of Large Numbers with a characterization of convergence rates, which is almost optimal, in the sense that it is arbitrarily close to the optimal rate for independent variables. Moreover, we prove an inequality comparing the joint distributions with the product distributions of the margins, similar to the well-known Newman's inequality for characteristic functions of associated variables. As a consequence, we prove a Central Limit Theorem together with its functional counterpart, and also the convergence of the empirical process for this class of weak dependent variables.
Theory of weak asymptotic autonomy of pullback stochastic weak attractors and its applications to 2D stochastic Euler equations driven by multiplicative noise
Ergodic Properties of weak Asymptotic Pseudotrajectories for Set-valued Dynamical Systems
描述诸如渐近自治微分方程或随机逼近过程等各种确定性和随机过程的渐近行为的一个成功方法是将其与适当选择的极限半流相关联。Benaïm和Schreiber（2000）定义了一类这样的随机过程，他们称之为弱渐近伪轨线，并研究了它们的遍历行为。特别是，他们证明了弱 * 极限点的经验措施与这些过程几乎肯定是不变的相关的确定性半流。Benaïm，Hofbauer and Sorin（2005）将这种方法推广到集值动力系统。我们追求的类比延伸到这些设置弱渐近伪轨线的遍历性。
